The sources thus far have mainly dealt with movement in shared spaces, but perhaps even more fundamental is the question of how much fits into a space and how best to utilize this limited resource. There have been a number of papers devoted to the problem of organizing storage units in real-world warehouses, although much of the prior work has focused on solving the various engineering challenges involved.
For example, Amato et al. [70] study control algorithms for warehouse robots, assuming a continuous distribution of item locations throughout the warehouse and ignoring the benefits of intelligent item placement. In a similar vein, Chang et al. [71] attempt to minimize unnecessary task repetition using genetic algorithms, thus shortening robot travel times, but assume a fixed storage scheme regardless of differing access frequencies. Sarrafzadeh and Maddila [72] use a discrete grid-based model, as is done in this work, but their focus is still an engineering one, concerned primarily with robot path-finding and constructing clearings through which to move. Closer to the work herein, Pang and Chan [73] address the question of where certain items should be stored in the warehouse, proposing a data-mining
approach to determine the relationships between products and co-locating those that are often purchased together. Experimental analysis shows that their methodology outperforms a simple greedy policy, but they do not present any formal proofs on the performance of their approach.
The word “warehouse” has been used for various optimization problems. In the context of operations research, the warehouse problem was proposed by Cahn [74] and later refined and extended by Charnes and Cooper [75] and Wolsey and Yaman [76].
This may sound related to the work done in this dissertation, but its focus is on the logistics of managing a warehouse’s stock in the face of changing demand. The word is also used in the context of coordinated motion planning under the name of the warehouseman’s problem. As mentioned in Section2.1.3, this is a multi-agent motion planning problem amidst obstacles. It has been shown to be PSPACE-hard [33,77], but efficient solutions exist for restricted versions (see, e.g., [31]).
While the approach used in this dissertation is theoretic in nature, the high complexity of the warehouseman’s problem is avoided by restricting the shapes of boxes (to unit squares) and the allowed layout of boxes (by introducing additional empty working space throughout to facilitate easy motion). The problems studied in Chapter 5 are less focused on motion planning and more on how to organize the warehouse’s contents to ensure efficient processing of a series of access requests.
More closely related to this work, however, is the dial-a-ride problem [78]. In this problem, a set of users must be conveyed from source locations to specified destinations in a metric space. The goal is to plan a route (or routes, in the case of multiple vehicles or the more general k-server problem [79]) that satisfies all
transportation requests while minimizing total distance traveled. One key difference is that the source locations are fully specified by the problem input, whereas in the warehouse problem presented in Chapter 5the location of requested boxes can be adjusted according to need, and how best to do so is central to the problem.
Additionally, while packing problems are well studied, both as static [80,81]
or online problems [82], the focus of this work is more on reducing retrieval times by organizing storage units based on their online access frequency and is similar in spirit to online algorithms for self-organizing memory structures [83,84]. Further details on this connection can be found in Chapter 5.
Chapter 3: On the Complexity of an Unregulated Traffic Crossing
3.1 Introduction
As autonomous and semi-autonomous vehicles become more prevalent, there is an emerging interest in algorithms for controlling and coordinating their motions to improve traffic flow. The steady development of motor vehicle technology will enable cars of the near future to assume an ever-increasing role in the decision making and control of the vehicle itself. In the foreseeable future, cars will have the ability to communicate with one another in order to better coordinate their motion.
This chapter considers two algorithmic formulations of a simple and fundamental geometric optimization problem involving coordinating the motion of vehicles through an intersection.
Traffic congestion is a complex and pervasive problem with significant economic ramifications, costing drivers in the United States over $305 billion in 2017 alone [85].
Practical engineering solutions will require consideration of myriad issues, including the physical limitations of vehicle motion and road conditions, the complexities and dynamics of traffic and urban navigation, external issues such as accidents and break-downs, and human factors. This chapter is focused on an algorithmic problem, called the traffic-crossing problem, that involves coordinating the motions of a set of vehicles moving through an intersection. In urban settings, road intersections are regulated by traffic lights or stop/yield signs. Like an asynchronous semaphore,
a traffic light locks the entire intersection preventing cross traffic from entering it, even when there is adequate space to do so. Some studies have proposed a less exclusive approach in which vehicles communicate either with one another or with a local controller that allows vehicles, possibly moving in different directions, to pass through the intersection simultaneously if it can be ascertained (perhaps with a small adjustment in velocities) that the motion is collision-free (see, e.g., [45]).
Even though such systems may be beyond the present-day automotive technology, the approach can be applied to controlling the motion of parcels and vehicles in automated warehouses [46].
Prior work on autonomous vehicle control has generally taken a high-level view (e.g., traffic routing [1–4]) or a low-level view (e.g., control theory, kinematics, etc.
[5,6]). Between these extremes, there has been a great deal of work on decentralized models of crowd motion, including methods based on velocity obstacles [47], reciprocal collision avoidance [49], and implicit crowds [86]. Much closer to the approach presented in this chapter is the work on autonomous intersection management (AIM) [45,59–62,87,88]. This work, however, largely focuses on the application of multi-agent techniques and deals with many real-world issues. As a consequence, formal complexity bounds are not proved.
Consider, instead, a simple problem formulation of the traffic-crossing problem, which captures the essential computational challenges of coordinating crosswise motion through an intersection. Vehicles are modeled as line segments moving monotonically along axis-parallel lines (traffic lanes) in the plane. Vehicles can alter their speed, subject to a maximum speed limit, but they cannot reverse direction
nor make turns. The objective is to plan the collision-free motion of these segments as they move to their goal positions.
After a formal definition of the traffic-crossing problem in Section3.2, three results are presented. First, Section 3.3 shows that this problem is NP-complete.
While this is a negative result, it shows that this problem is of a lower complexity class than similar PSPACE-complete motion-planning problems, like sliding-block problems [33]. Second, in Section3.5 a constrained version is considered in which vehicles travel vertically at a fixed speed. This variant is motivated by a scenario in which traffic moving along one axis (e.g., a major highway) has priority over crossing traffic (e.g., a small road). An algorithm based on plane-sweep is presented that solves this problem in O(n log n) time. Finally, the problem is considered in a discrete setting in Section 3.6, which simplifies the description of the algorithms while still capturing many of the interesting scheduling elements of the problem.
As part of this consideration, a solution to the problem is provided that limits the maximum delay of any vehicle and it is proven that this solution is asymptotically optimal.